How to solve rational expressions (the 20-second method)

• Simplify: Factor numerator & denominator → cancel common factors → list excluded values.
• Add/Subtract: Factor denominators → find LCD → rewrite each fraction → combine → simplify.
• Multiply: Factor → cancel first → multiply across → simplify.
• Divide: Multiply by reciprocal → factor & cancel → multiply → simplify.

If your answer looks “too simple,” check whether you canceled something that was part of a sum/difference.

Rational Expressions: what they are (and why they matter)

A rational expression is a fraction where the numerator and/or denominator are polynomials. Examples include (x² − 1)/(x − 1), (2x + 3)/(x² + 5x + 6), or even (x − 4)/7 (a polynomial over a constant is still rational). Rational expressions show up everywhere: algebra homework, function graphs with asymptotes, calculus limits, and many real-life formulas that look like “something divided by something.”

The #1 rule that makes rational expressions different from normal fractions is the domain restriction: you can’t divide by zero. That means any value of x (or any variable) that makes the denominator equal to zero is not allowed. Our calculator automatically finds these “excluded values” so you can simplify safely and avoid accidental mistakes.

• Simplifying a rational expression means factoring and canceling common factors (when allowed).
• Adding/subtracting rational expressions requires a common denominator (often the LCD).
• Multiplying/dividing is usually faster: factor, cancel, then multiply.
• Domain restrictions come from the original denominators (and any extra denominators you introduce).

The “fraction rules” — upgraded for algebra

Think of rational expressions like normal fractions, but with polynomials. The familiar rules still apply: reduce by canceling common factors, and use common denominators when adding or subtracting. The difference is that you usually need to factor first.

If you have f(x) = P(x)/Q(x), the simplified form is found by factoring:

• Factor numerator: P(x) = (common)(rest)
• Factor denominator: Q(x) = (common)(rest)
• Cancel the common factor (but keep the excluded values from the original denominator).

Example: (x² − 1)/(x − 1). Factor the numerator: x² − 1 = (x − 1)(x + 1). Then: ((x − 1)(x + 1))/(x − 1) = x + 1, but with a restriction: x ≠ 1.

To add a/b + c/d, you use (ad + bc)/bd. For rational expressions, the idea is identical, except b and d are polynomials, and the best approach is the least common denominator (LCD).

Example: 1/(x − 1) + 1/(x + 1). The LCD is (x − 1)(x + 1). Rewrite: (x + 1)/((x − 1)(x + 1)) + (x − 1)/((x − 1)(x + 1)) so the sum is (2x)/((x − 1)(x + 1)), then simplify if possible.

Multiplication is straightforward: (P/Q) × (R/S) = (P·R)/(Q·S). Division means “multiply by the reciprocal”: (P/Q) ÷ (R/S) = (P/Q) × (S/R). In both cases, factor and cancel first to keep numbers small and steps clean.

Worked examples you can copy into the calculator

Input: (x^2-9)/(x^2-3x)
Factor: numerator (x − 3)(x + 3), denominator x(x − 3)
Cancel: (x + 3)/x with restriction x ≠ 0, 3.

Input: 2/(x-2) + 3/(x+2)
LCD: (x − 2)(x + 2)
Combine: 2(x+2) + 3(x−2) over the LCD = (5x − 2)/((x − 2)(x + 2))
Restrictions: x ≠ 2, −2.

Input: (x^2-1)/(x^2-4) × (x-2)/(x-1)
Factor: (x−1)(x+1)/(x−2)(x+2) × (x−2)/(x−1)
Cancel: (x+1)/(x+2)
Restrictions: from original denominators x ≠ 2, −2, 1.

Input: (x/(x-1)) ÷ ((x+2)/x)
Multiply by reciprocal: (x/(x−1)) × (x/(x+2)) = x^2/((x−1)(x+2))
Restrictions: x ≠ 0, 1, −2. (Because you can’t divide by zero and the divisor can’t be zero.)

Tip: Use parentheses around numerators/denominators like (x^2-1)/(x-1). Use ^ for exponents (x^2). Multiplication can be written as * or implied (2x).

Frequently Asked Questions

• Why do I get “x ≠ …” excluded values?

  Those values make a denominator equal to zero. Rational expressions are undefined there, even if a factor cancels during simplification. The simplified form may look simpler, but the original restrictions still apply.

• Is canceling the same as plugging in?

  No. Canceling is an algebraic simplification that works for all allowed values. Plugging in a value is evaluation, and it can fail if the value is excluded.

• What if my expression uses a different variable (like “t”)?

  Use the Variable box (default is x) and set it to t, y, or whatever you’re using. Excluded values will be solved for that variable.

• Does simplifying change the function?

  It changes the formula but not the function’s values on the allowed domain. The only difference is at the excluded points (holes). That’s why we show restrictions—so the math stays honest.

• What input format works best?

  Use parentheses around numerators/denominators, write exponents with ^, and avoid “pretty” symbols. Example: (2x+3)/(x^2+5x+6).

• Can this solve completely automatically like a CAS?

  This calculator focuses on the most common rational-expression operations: simplify, add/subtract, multiply/divide, and domain restrictions. For advanced graphing, use our Graphing Calculator link.

Educational tool only. Always verify steps required by your class (some teachers require factoring shown).